The generalized Pareto distribution (GPD) is often used in the statistical analysis of climate extremes. For a sample of independent and identically distributed observations, the parameters of the GPD can be estimated by the maximum likelihood (ML) method. In this paper, we drop the assumption of identically distributed random variables. We consider independent observations from GPD distributions having a common shape parameter but possibly an increasing trend in the scale parameter. Such a model, with increasing scale parameter, can be used to describe a trend in the observed extremes as time progresses.

Estimating an increasing trend in a distribution parameter is common in the field of isotonic regression. We use ideas and tools from that area to compute ML estimates of the GPD parameters. In a simulation experiment, we show that the iterative convex minorant (ICM) algorithm is much faster than the projected gradient (PG) algorithm.

We apply the approach to the daily maxima of the central England temperature (CET) data. A clear positive trend in the GPD scale parameter is found, leading to an increase in the 100-year return level from about 31º in the 1880s to 34º in 2015.